IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 11, NO. 2, FEBRUARY 1999 209 Nonlinear Finite-Element Semivectorial Propagation Method for Three-Dimensional Optical Waveguides A. Cucinotta, S. Selleri, and L. Vincetti Abstract—A nonlinear semivectorial beam propagation method based on the finite element method is presented. It applies to three-dimensional optical waveguides and accounts for polariza- tion effects. To show the usefulness of this approach numerical results regarding both (quasi-TE) and (quasi-TM) waves are presented for directional couplers with nonlinear cores. The limits of scalar formulations are highlighted especially when working near and above the coupler threshold power. Index Terms—Finite-element method, integrated optics, nonlin- ear optical propagation, semivectorial beam propagation method. I. INTRODUCTION A T PRESENT, integrated optics has reached a consid- erable maturity toward the fulfilment of devices for all-optical signal computing and processing. Basic devices as logical gates, switches, splitters, and amplifiers, have been in- tegrated and a huge variety of functionalities demonstrated. An accurate analysis of these devices is thus mandatory in order to improve their performances. Due to the high index step, typical of integrated optics materials, and to the permittivity variation along the transverse directions, the electromagnetic waves which propagate in these devices present hybrid field distributions and the behavior of the different polarizations ( or quasi-TE and or quasi-TM modes) can significantly change. These differences considerably increase when strongly nonlinear index variations are considered. Analysis and design based on scalar approaches can thus provide results that significantly differ from those obtained by means of vectorial approaches both qualitatively and quantitatively [1]–[3]. In spite of this, in the last years, many propagation methods which solve the scalar wave equation have been developed for both second- and third-order nonlinear media [4]–[7]. In this letter a semivectorial formulation is presented as first attempt to overcome the limits of scalar approaches. It has been derived from a full-vectorial beam propagation method (BPM) based on finite elements (FE’s) [8] and has been applied to the analysis of nonlinear devices. The orig- inal FE-BPM [8] propagates the three components of the magnetic field. In the present approach only two of them are considered; one of the transverse components, according to the studied polarization, and the longitudinal one. The divergence condition is included in the formulation. The obtained approach can not be defined as a full vectorial Manuscript received September 3, 1998; revised October 27, 1998. The authors are with the Dipartimento di Ingegneria dell’Informazione, University of Parma, I-43100 Parma, Italy. Publisher Item Identifier S 1041-1135(99)01085-X. approach as one of the transverse components is neglected. The authors will refer to the presented formulation as a semivectorial one even if this term is sometimes used in literature for formulations including just one component, that along or according to the polarization to be investigated [9]. FE-scalar formulations, which propagate one component, can be derived from the proposed semivectorial approach and have been already presented in [10]. The main advantage of the proposed semivectorial formu- lation is that it is simpler and faster than vectorial ones as it produces a system matrix with dimension equal being the numerical unknowns, rather than the very time consuming as in the vectorial case. Nevertheless, it is still able to distinguish the field polarizations and produces results considerably more accurate than scalar approaches. The proposed formulation and the procedure used to ap- proach the nonlinear problem are presented in the next section. Numerical results concerning nonlinear directional couplers are reported in Section III showing the performance of the semivectorial FE-BPM and highlighting the limits of scalar approaches. Conclusions follow. II. FORMULATION Consider the curl–curl equation in term of the magnetic field (1) where is the wavenumber in the vacuum (2) is the nonlinear relative permittivity and the electric field; is the linear relative permittivity and , measured in , is related to the nonlinear Kerr coefficient through and being the light speed and the per- mittivity in vacuum, respectively [11]. The coefficient has units of . By expressing , with a phase factor and the propagation direction and by applying the standard Galerkin method to (1) it yields [8] (3) 1041–1135/99$10.00  1999 IEEE